Answer to Question #142497 in Complex Analysis for Doll

Question #142497
Prove that u(x,y) given by the following is harmonic obtain it's corresponding conjugate and original function f(z)
u(x,y)=x^2-y^2
1
Expert's answer
2020-11-05T16:51:19-0500

"\\displaystyle u(x, y) = x^2 - y^2\\\\\n\n\\frac{\\partial u}{\\partial x} = 2x, \\frac{\\partial^2 u}{\\partial x^2} = 2\\\\\n\n\n\\frac{\\partial u}{\\partial y} = -2y, \\frac{\\partial^2 u}{\\partial y^2} = -2\\\\\n\n\n\\textsf{Since}\\, \\frac{\\partial^2 u}{\\partial x^2} + \\frac{\\partial^2 u}{\\partial y^2} = 0, \\, u\\, \\textsf{is harmonic}\\\\\n\n\n\\textsf{Since a harmonic function is analytic}\\\\\n\n\n\n\\frac{\\partial u}{\\partial x} = \\frac{\\partial v}{\\partial y}\\\\\n\n\n2x = \\frac{\\partial v}{\\partial y}\\\\\n\nv = \\int 2x\\, \\mathrm{d}y\\\\\n\n\nv = 2xy + C\\\\\n\n\n\n\\therefore \\textsf{The complex conjugate is}\\, 2xy + C\\\\\n\n\n\\therefore\\, f(z) = u + jv = (x^2 - y^2) + j2xy + C= (x + jy)^2 + C = z^2 + C\\\\\n\n\n\\textsf{Where}\\, C \\, \\textsf{is an arbitrary constant obtained}\\\\\\textsf{from the integration process}"


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